18 problems

Let $n$ be the number of vertices of a unicyclic graph, let $C_n$ denote the cycle on $n$ vertices, and let $P_n^6$ denote the unicyclic graph specified in the source. Caporossi et…

Let $\pi=(d_0,d_1,\ldots,d_{k-1},1,\ldots,1)$ be a graphic unicyclic degree sequence with $2\le d_0\le d_1\le\ldots\le d_{k-1}$ and $d_k=\ldots=d_{n-1}=1$. Let $\mathcal{U}_{\pi}$…

Akbari–Kumar–Mohar–Pragada–Zhang conjecture. If $k\equiv 3\pmod 4$, then $s^+(G)>n>s^-(G)$; if $k\equiv 1\pmod 4$, then $s^+(G)<n<s^-(G)$. This predicts that the residue of the odd…

Let $T$ be a tree serving as a reservoir, let $u,v$ be vertices of $T$, and fix a rule for adding a closure edge between $u$ and $v$ to form a unicyclic graph. Consider the minimal…

Let $G$ be a unicyclic graph of order $n$, and let $k$ be the odd length of its unique cycle. Unicyclic graph square-energy conjecture. … The conjecture is motivated by an exact ca…

Let $G$ be a unicyclic graph on $n$ vertices, and let $\mathfrak{c} \in \mathbb{Z}_{>0}^n$. Write $\operatorname{Ker}(\pi_G^{\mathfrak{c}})$ for the associated toric ideal, and cal…

Let $G_1$ and $G_2$ be connected unicyclic graphs, where either each has a 3-cycle and an odd number of vertices, or each has a $c$-cycle for some $c\geq4$. Let $\mathbf{X}_G$ deno…

Let $D=(d_1,d_2,\dots,d_{t-1},d_t,1,\dots,1)$ be a degree sequence with $d_t\geq 2$ and $d_2\neq 2$, and let $G$ be a unicyclic graph with degree sequence $D$. Define … and let…

Regularity conjecture for t-path ideals. The displayed formula should hold. In particular,

Let $G$ be a unicyclic graph, let $\Delta$ be its distance squared matrix, let $\ell$ be the number of leaves of $G$, and let $q$ be the number of negative eigenvalues contributed…

Let $G$ be a unicyclic graph of order $n$ whose cycle has size $k$. The path energy $PE(G)$ is defined for such graphs. Path energy conjecture for unicyclic graphs. $PE(G)$ depends…

A simple $n$-vertex graph has a prime vertex labeling if its vertices can be injectively labeled with the integers $1,2,3,\ldots,n$ such that adjacent vertices have relatively prim…

Let $C_n\star P_2\star S_m$ denote the cycle pendant star formed from a cycle, a path, and a star. A graph is prime if it has a vertex labeling by consecutive integers starting at…

Let a unicyclic graph be a graph containing exactly one cycle as a subgraph. A graph is prime if its vertices can be injectively labeled with the integers $1,2,\ldots,n$ so that ad…

Conjecture 1. For $n\geq 5$,

Modified Ilić and Ilić conjecture. (1) For $G\in\mathcal{U}_{n,l}^g$, if more than two vertices on the cycle have degree at least $3$, then $U^0\preceq G$, with equality if and onl…

Ilić and Ilić's conjecture. Among all $n$-vertex unicyclic graphs, the graph $U_{n,l}^{3,0}$ has the minimum Laplacian coefficients $c_k$, $k=0,\dots,n$; equivalently,…

Let $G$ be a unicyclic graph and let $T$ be a spanning tree of $G$. The spanning-tree upper-bound conjecture. If $T$ is a spanning tree of a unicyclic graph $G$, then … This conjec…